Problem: Rewrite as a simplified fraction. ${0.5\overline{90} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 590.909...\\ 10x &= 5.909...\end{align*}}$ Subtracting the two equations, we have: ${990x = 585}$ ${x = \dfrac{585}{990}} $ ${x = \dfrac{13}{22}} $